---
book: 10
number: 49
id: "X.49"
kind: "construction"
uses: ["[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-36]]", "[[book-5/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.49"
license: "CC-BY-SA-4.0"
---

# X.49

*To find the second binomial straight line*.

## Proof

Let two numbers *AC*, *CB* be set out such that the sum of them *AB* has to *BC* the ratio which a square number has to a square number, but has not to *AC* the ratio which a square number has to a square number; let a rational straight line *D* be set out, and let *EF* be commensurable in length with *D*; therefore *EF* is rational.

Let it be contrived then that, as the number *CA* is to *AB*, so also is the square on *EF* to the square on *FG*; [[book-10/proposition-6|X. 6, Por.]] therefore the square on *EF* is commensurable with the square on *FG*. [[book-10/proposition-6|X. 6]]

Therefore *FG* is also rational.

Now, since the number *CA* has not to *AB* the ratio which a square number has to a square number, neither has the square on *EF* to the square on *FG* the ratio which a square number has to a square number.

Therefore *EF* is incommensurable in length with *FG*; [[book-10/proposition-9|X. 9]] therefore *EF*, *FG* are rational straight lines commensurable in square only; therefore *EG* is binomial. [[book-10/proposition-36|X. 36]]

It is next to be proved that it is also a second binomial straight line.

For since, inversely, as the number *BA* is to *AC*, so is the square on *GF* to the square on *FE*, while *BA* is greater than *AC*, therefore the square on *GF* is greater than the square on *FE*.

Let the squares on *EF*, *H* be equal to the square on *GF*; therefore, convertendo, as *AB* is to *BC*, so is the square on *FG* to the square on *H*. [[book-5/proposition-19|V. 19, Por.]]

But *AB* has to *BC* the ratio which a square number has to a square number; therefore the square on *FG* also has to the square on *H* the ratio which a square number has to a square number.

Therefore *FG* is commensurable in length with *H*; [[book-10/proposition-9|X. 9]] so that the square on *FG* is greater than the square on *FE* by the square on a straight line commensurable with *FG*.

And *FG*, *FE* are rational straight lines commensurable in square only, and *EF*, the lesser term, is commensurable in length with the rational straight line *D* set out.

Therefore *EG* is a second binomial straight line. Q. E. D.